One guitarist causes an oscillation given by $$y_1(t)=A\sin({\omega}t+\phi_1)$$
Another guitarist causes an oscillation given by $$y_2(t)=A\sin({(\omega+\delta)}t+\phi_2)$$
Furthermore, $$y(t)=y_1(t)+y_2(t)$$
Given formula (1) $$e^{ia}+e^{ib}=2e^{i\frac{(a+b)}{2}}\cos(\frac{a-b}{2})$$
Formula (1) should be used to show $$y(t)=2A\cos(\frac{\delta}{2}t-\frac{\phi_1 -\phi_2}{2})\sin((\omega+\frac{\delta}{2})t+\frac{\phi_1 +\phi_2}{2})$$ I've attempted adding $y_1(t)$ and $y_2(t)$, hoping that something useful would drop out. However, this becomes quite messy after using angle sum identities and I can't make sense of it. I've considered double angle formulae, product-to-sum, sum-to-product formulae. What is a good approach to solving this problem?
This doesn't use your equation (1), but by the Prosthaphaeresis Formulas (see http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html),
$$\begin{aligned}&\sin(\omega t + \phi_1)+\sin((\omega+\delta)t+\phi_2) = \\ &=2\sin\left(\frac{1}{2}\left(\omega t + \phi_1 + (\omega+\delta)t+\phi_2\right)\right)\cos\left(\frac{1}{2}\left(\omega t + \phi_1 - (\omega+\delta)t-\phi_2\right)\right)\\ &=2\sin\left((\omega+\frac{\delta}{2})t+\frac{\phi_1+\phi_2}{2}\right)\cos\left(-\frac{\delta}{2}t+\frac{\phi_1-\phi_2}{2}\right)\\ &=2\sin\left((\omega+\frac{\delta}{2})t+\frac{\phi_1+\phi_2}{2}\right)\cos\left(\frac{\delta}{2}t-\frac{\phi_1-\phi_2}{2}\right) \end{aligned}$$